Open Access Austin Journal of Surgery

Review Article Fibonacci Series, Golden Proportions, and the Human Biology

Dharam Persaud-Sharma* and James P O’Leary Florida International University, Herbert Wertheim Abstract College of Medicine Pythagoras, Plato and Euclid’s paved the way for Classical Geometry. *Corresponding author: Dharam Persaud-Sharma, The idea of shapes that can be mathematically defined by equations led to the Florida International University, Herbert Wertheim creation of great structures of modern and ancient civilizations, and milestones College of Medicine, Miami, FL, 33199, USA, Email: in mathematics and science. However, classical geometry fails to explain the [email protected] complexity of non-linear shapes replete in nature such as the curvature of a flower or the wings of a Butterfly. Such non-linearity can be explained by Received: April 01, 2015; Accepted: June 25, 2015; fractal geometry which creates shapes that emulate those found in nature with Published: July 02, 2015 remarkable accuracy. Such phenomenon begs the question of architectural origin for biological existence within the universe. While the concept of a unifying equation of life has yet to be discovered, the Fibonacci sequence may establish an origin for such a development. The observation of the Fibonacci sequence is existent in almost all aspects of life ranging from the leaves of a fern tree, architecture, and even paintings, makes it highly unlikely to be a stochastic phenomenon. Despite its wide-spread occurrence and existence, the Fibonacci series and the Rule of Golden Proportions has not been widely documented in the human body. This paper serves to review the observed documentation of the Fibonacci sequence in the human body.

Keywords: Fibonacci; Surgery; Medicine; Anatomy; Golden proportions; Math; Biology

Introduction phenomenon known as the Fibonacci sequence and the rule of ‘Golden Proportions’ may serve as a starting point for uncovering the Classical geometry includes the traditional shapes of triangles, methods of a universal architect, should one exist. squares, and rectangles that have been well established by the brilliance of Pythagoras, Plato, and Euclid. These are fundamental Fibonacci and rule of golden proportions shapes that have forged the great structures of ancient and modern Fibonacci sequences: The Fibonacci sequence was first day civilization. However, classical geometry fails to translate into recognized by the Indian Mathematician, Pingala (300-200 B.C.E.) complex non-linear forms that are observed in nature. To explain in his published book called the Chandaśāstra, in which he studied such non-linear shapes, the idea of fractal geometry is proposed which grammar and the combination of long and short sounding vowels states that such fractal shapes/images possess self-similarity and [1,2]. This was originally known as mātrāmeru, although it is now can be of non-integer or non-whole number dimensions. Whereas known as the Gopala-Hemachandra Number in the East, and the classical geometric shapes are defined by equations and mostly whole Fibonacci sequence in the West [1,2]. Leonardo Pisano developed number (integer) values, the shapes of fractal geometry can be created the groundwork for what is now known as “Fibonacci sequences” to by iterations of independent functions. Observing the patterns the Western world during his studies of the Hindu-Arab numerical created by repeating ‘fractal images’ closely emulate those observed system. He published the groundwork of Fibonacci sequences in in nature, thus questioning whether this truly is the mechanism of his book called Liber Abaci (1202) in Italian, which translates into origination of life on the planet. Two important properties of fractals English as the “Book of Calculations”. However, it should be noted include self-similarity and non-integer or non-whole number values. that the actual term “Fibonacci Sequences” was a tributary to The idea of ‘self-identity’ is that its basic pattern or fractal is the same Leonardo Pisano, by French Mathematician, Edouard Lucas in 1877 at all dimensions and that its repetition can theoretically continue [3]. The Fibonacci sequence itself is simple to follow. It proposes that to infinity. Examples of such repeating patterns at microscopic and for the integer sequence starting with 0 or 1, the sequential number is macroscopic scales can be seen in all aspects of nature from the the sum of the two preceding numbers as in Figure 1. florescence of a sunflower, the snow-capped peaks of the Himalayan Mountains, to the bones of the human body. A) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…. ∞ The observation of self-similarity belonging to fractal geometry found in multiple aspects of nature, leads to the question of whether Or such an occurrence is merely stochastic or whether it has a functional B) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… ∞ purpose. While a singular unifying equation to define the creation Figure 1: Illustration of Fibonacci Sequences. A) Fibonacci Series starting and design of life has yet to be determined, an un-coincidental with 1 B) Fibonacci Series starting with 0.

Austin J Surg - Volume 2 Issue 5 - 2015 Citation: Persaud-Sharma D and O’Leary JP. Fibonacci Series, Golden Proportions, and the Human Biology. ISSN : 2381-9030 | www.austinpublishinggroup.com Austin J Surg. 2015;2(5): 1066. Persaud-Sharma et al. © All rights are reserved Dharam Persaud-Sharma Austin Publishing Group

Table 1: Ratios of Fibonacci Numbers for the sequences starting with 1. Ratios approximate Fibonacci value of Phi (φ) 1.618.

FN + 1 FN (FN + 1)/FN 2 1 2

3 2 1.5000

5 3 1.6666

8 5 1.6000

13 8 1.6250

21 13 1.6154

34 21 1.6190

55 34 1.6176

89 55 1.6182 Figure 2: Golden Rectangle and Golden Spiral. 1.6179 144 89 (Nearest approximation of φ)

As shown in Figure 1, Fibonacci sequences can either begin with 0 or 1. Although the sequence, originally proposed in Liber Abaci by Leonardo (follows the sequence in A) begins with 1, this sequence essentially reduces to the recurrence relationship of: FN = Fn-1 + Fn-2 Given that the starting numbers of the sequence can be either:

A) F1 = 1 F2 = 1 or B) F1 = 0 F2 = 1.In this sequence it can be seen that each additional number after the given conditions of 1 or 0, is a direct result of the sum of the two preceding numbers. For example, in series (A), 1+ 1 = 2, where 2 is the third number in the sequence. Figure 3: Golden Spiral in Nature.

To find the third number (F4), simply add F3 = 2 and F2 = 1, which equals 3 (F4). Golden proportion, rectangle, and spiral: Another observation that can be made from the series of the Fibonacci numbers includes the rule of golden proportions. In essence, this is an observation that the ratio of any two sequential Fibonacci numbers approximates to the value of 1.618, which is most commonly represented by the Greek Letter Phi (φ). The larger the consecutive numbers in the sequence, the more accurate the approximation of 1.618. A summary of these results illustrating this concept can be seen in Table 1. Furthering this observation, shapes can be created based upon length measurements of the Fibonacci numbers in sequence. For example, creating a rectangle with the values of any of the two successive Fibonacci numbers form what is known as the “Golden Rectangle”. These rectangles can be divided into squares that are equally sided and are of smaller values from the Fibonacci sequence Figure 2. Figure 4: The human hand superimposed on the Golden Rectangle, with approximations to the Golden Spiral [4]. Rectangles created using consecutive numbers from the Fibonacci sequence can be divided into equally sided squares of such numbers Human anatomy and Fibonacci as well. The length of a single side can be divided into smaller values Proportion of human bones: In the human body, there are of the Fibonacci sequence. Connecting the diagonals of each of the instances of the Fibonacci Phi (φ) although it has not been widely squares creates a spiral, known as the Golden Spiral. discussed. In 1973, Dr. William Littler proposed that by making a This image compares the spiral of the human ear, the shell of a clenched fist, Fibonacci’s spiral can be approximated in Figure 4 [4]. nautilus, and the golden spiral. Both the human ear and the shell of a In this paper, Litter proposed that such geometry is observed nautilus approximate the dimensions of the golden spiral. based upon ratio of the lengths of the phalanges, and that the flexor Connecting the corners of each of the squares by an arc reveals a and extensor movement of the primary fingers approximate the spiral pattern. This spiral pattern is known as the ‘golden spiral’. This golden spiral. Such a spiral would have to be created based upon the pattern is seen in several different forms in nature including, but not relationship between the metacarpophalangeal and interphalanges of limited to the human ear and the shell of a nautilus as seen in Figure 3. the digits [4]. However, due a lack of statistical and well documented

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have attempted to quantitatively characterize the parameters of an aesthetically appealing smile. The most logical starting point is that of the rule of golden proportions. In 1973, Lombardi was the first to officially propose the existence of having proportionate teeth, but dismissed the idea of using the rule of golden proportions to create aesthetic teeth [9]. He did suggest an underlying repeated ratio of the maxillary anterior teeth for aesthetics, but stated the inappropriate use of the rule of golden proportions for such purposes. In 1978, Levin was the first to observe that: 1) the width of the maxillary central incisor is in golden proportion to the width of the lateral incisor. 2) The width of the maxillary lateral incisor is in golden proportion to the width of the canine [10]. Based upon this observation, Levin later developed a tooth caliper to gauge whether an individual’s teeth are in golden proportion to one another, and diagnostic grid to verify if the teeth are appropriately spaced. Further research into the relationship between the golden proportion, dental arrangement, and an aesthetic smile was developed by Rickets in 1982. He implemented Figure 5: Phalangeal lengths as reported by Hamilton et al. [6]. For the index, the use of the golden proportions in the treatment of patients [11,12]. second and third digits: ratios of the DIP-tip/PIP-DIP/MCP-PIP distances While Levin’s observation of the golden ratio existing in dentition is were 1:1.3:2.3, and 1:1:2 for the little finger. DIP: Distal interphalangeal; PIP: Proximal Interphalangeal; MCP: Metacarpophalangeal Joint. undeniable, its application to developing an aesthetically appealing smile has recently been questioned and often dismissed [9,13-15]. empirical data, accurate representations of the golden spiral could In 2007, Dio et al. attempted to empirically define the basis for not be readily ascertained. In 1998, Gupta et al., confirmed that the determining/judging beauty [16]. Dio et al., utilized fMRI imaging of motion of the phalanges do approximate the pattern of the golden the brain to formulate an association between brain stimulation from spiral, excluding for the fifth digit because of abduction of the digit the interpretation of visual models with different physical proportions- during extension [5]. Through empirical evidence obtained through using a model constructed according to golden proportions as an radiographic analysis of the hands of 197 individuals, Hamilton and independent variable. The results of the study indicate that defining Dunsmuirin 2002 concluded that the ratios between the phalange beauty is a joint process of cortical neurons, where Dio et al. identified lengths of the digits do not approximate the golden ratio value of that objective beauty stimulates the insula, whereas subjective beauty (φ) 1.618 [6]. However, Hutchison and Hutchison (2010) showed based upon the judgment of the individual perceiving the experience that the data collected by Hamilton et al. (2002) overlooked the shows functional excitation of the Amygdala. Dio et al. echo the relationship that Littler actually proposed. Hamilton and Dunsmuir sentiments referenced by Gombrich [17] and Ramachandran [18], in (2010) confirmed that the phalangeal length ratio data obtained that the criteria for making overall assessments of beauty seem to be from their subjects compared to those that were almost arbitrarily related to our biological heritage, although it cannot be explained by a listed by Littler in 1973, were in fact comparable and approximated conscious explanation [16]. In our opinion, while the existence of the the Fibonacci value of (φ) of 1.618. What Hamilton et al. failed to golden ratio continues to be revealed in several different instances of notice in their own data was that the values they obtained from their the human anatomy, attempting to use its proportions as a measure of research approximated a Lucas series, which essentially underscores beauty is not something that should be standardized. It is influenced a Fibonacci sequence in that the sum of the first two lengths equaled by many different factors including cultural influences. This is the third length [7]. something that may have been overlooked by Dio et al. study, in that Hamilton and Dunsmuir’s findings highlighted that the index, in the demographic information about the 14 participants in the study second, and third digits follow a series of 1x, 1.3x, 2.3x [7]. Hamilton was not included. The idea of ‘beauty’ itself is a subjective term and et al. provided length measurement data for the fourth digit (little not one that can be universally held constant. The diversity of biology finger) that follows the initial values of a Fibonacci sequence of in its essence is what constitutes its beauty, while the observation of 0,1,1,2, represented by y, y, and 2y in Figure 5 [7]. Hence, the data the golden proportion in nature highlights an architectural design, collected by Hamilton et al., supported Littler’s initial proposal of proportions, and symmetry. a clenched fist approximating the dimensions of the golden spiral. Phyllotaxis, coronary arteries, and the human heart beat: Additionally, Hamilton’s data showed that the ratios for the length of Although the blood vessels of the human body and a plant have the phalanges follows the additive rule of a Lucas series for the index, different physiological components, they are very similar in their second and third digits, while approximating Fibonacci values for the basic function and visual appearance Figure 6. The vascular network fourth digit (little finger). in its basic structure is nearly identical to shapes that can be created by The aesthetics of a smile and beauty: The rule of golden fractal geometry, as seen in Figure 6C. Mathematically, the properties proportions has been proposed in an attempt to define anatomical of these structures are created by the self-similarity of the fractal. This beauty. It is commonly accepted that facial beauty is correlates with means that the whole image is the result of multiple smaller copies anatomical symmetry [8]. One of the main features of the human face of itself that share the same statistical properties at different scales of is the mouth and teeth. Thus, professionals within the field of dentistry measurement.

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Istockphotos.comA B C

cw.phys.ncku.edu.tw Osirix Hameroﬀ, S., Ultimate Computing, 1987

Figure 6: Comparison between the silhouettes of a tree (A), the blood vessels of a heart (B), and a fractal image (C).

Table 2: Table summarizing the number of branching from blood vessels of the Heart showing that branching numbers follow a Fibonacci Series [19]. Number of branches

2 LCA RCA

3 LAD LCx RCA

5 LAD LCx RI AM PDA

8 LAD Ldiag1 LCx OM1 Sep1 PDA Rdiag1 AM

13 LAD PLV1 PRV1 Sep3 Sep4 OM2 LDiag2 Sep2 OM3 PLV2 PLV3 PRV2 PRV3

Sequence branding of the coronary arterial tree. LCA: Left Coronary Artery; RCA: Right Coronary Artery; LAD: Left Anterior Descending; LCX: Left Circumflex; OM: Obtuse Marginal; AM: Acute Marginal; PDA: Posterior Descenting Artery; PRV: Posterior Right Ventricular (1) = lateral; (2) = intermediate; (3) = medial, Sep: Septal; RI: Ramus Intermedius (authors interpretation). Of importance here is that in 1977, Mitchison revealed that past 25 years, this work has mainly been led by French independent- the phyllotaxis of plants follows the Fibonacci sequence 1, 1, 2, 3, researcher Jean-Claude Perez. In 1991, Perez proposed that the DNA 5, 8, 13, 21… as seen in Figure 1. Likewise in 2011, Ashrafian and gene-coding region sequences were strongly related to the Golden Atasiounoted that the number of branches from vessels in the Ratio and Fibonacci/Lucas integer numbers [23]. His work was heart follows the Fibonacci sequences Table 2 [19]. Ashrafian and bolstered by Yamagishi et al who found consistency of a Fibonacci Atanasiou, supported their observation by correlating studies which series level of organization across the whole human genome [24]. In showed that atherosclerotic lesions in coronary arteries follow a 2010, Perez provided evidence that the genetic code in Table 3 not Fibonacci distribution [20]. only maps codons to amino acids, but also serves as a global error- detection scheme [25]. In 2013, Yetkin et al. showed that the golden ratio of Phi (φ) 1.618, exists within the cardiac cycle of the human heart beat. It is In the simplest explanation of his research, Perez analyzed the known that the time periods for the systolic and diastolic phases entirety of the human genome yet only evaluated a single strand vary with the method of measurement. The three most commonly considering that 1 strand is a mere complement of the other. As used techniques for determining systolic and diastolic phases of outlined by Perez in 2010, the cumulative number of each of the the cardiac cycleinclude: electrographical, echocardiogaphical, and 64 genetic codons for the three different codon reading frames was phonocardiographical techniques. Yetkin et al. defined systole as obtained [25]. Then analysis on 24 human chromosomes, using the the time between the ‘R wave’ and the end of the ‘T wave’ using an same method, was performed and the results were totaled for the electrocardiogram [21]. The data from their research included an 3 codon reading frames and the 24 chromosomes. The 64 codon assessment of 162 healthy subjects, and concluded that the ratio of population was arranged according the Genetic Code as seen in Table the Diastolic time interval/Systolic time interval was 1.611, while the 3. The table was then separated according to the 6 binary splits of a R-R interval/Diastolic time interval was 1.618 which is the value of Dragon Curve. A dragon curve is an irregularly shaped curve with a Phi (φ) [21]. self-similar pattern, hence a fractal image as shown in Figure 7. Fibonacci, Fractals, and the Human Genome: The supposition Conceptually, a Dragon Curve can be created by folding of a piece that the structure of DNA and its organization pattern is a fractal of paper into a long strip. Then as the strip of paper is unfolded, the was first proposed by Benoit Mandelbot in 1982 [22]. Mandelbot, adjacent segments are rotated and formed at right angles [26,27]. for the most part, is responsible for first coining the term “fractal” Computer modeling of this concept is shown in Figure 7. Perez’s and providing mathematical explanation non-euclidean geometries research in 2010makes several conclusions regarding the fractal observed in nature. In the early 1990’s, based upon the supposition nature of DNA. First, the universal genetic code serves as a macro- made by Mandelbot, it was proven that the Human genome contains level structural matrix [25]. This matrix serves as a regulatory fractal behavior - Fibonacci series, and the golden proportion. For the mechanism, controlling and balancing the codon population within

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Table 3: The 64 codon populations of the whole human genome for the 3 codon reading frames of single stranded DNA (2843411612 codons). In this table, the 3 values in each cell are: the codon label, the codon’s total population, the “Codon Frequency Ratio” (CFR). CFR is computed as: codon population x 64 / 2.843.411.612. (Where 2.843.411.612 is the whole genome cumulated codons). Then, if CFR < 1, the codon is rare, If CFR>1, the codon is frequent.

SECOND NUCLEOTIDE

T C A G TTT 109591342 TCT 62964964 TAT 58718182 TGT 57468177 T 2.4667 1.4172 1.3216 1.2935 TTC 56120623 TCC 43850042 TAC 32272009 TGC 40949883 C 1.2632 0.9870 0.7264 0.9217 T TTA 59263408 TCA 55697529 TAA 59167883 TGA 55709222 A 1.3339 1.2536 1.3318 1.2539 TTG 54004116 TCG 6263386 TAG 36718434 TGG 52453369 G 1.2133 0.1410 0.8265 1.1806 CTT 36828780 CTT 50494519 CAT 52236743 CGT 7137644 T 1.2791 1.1365 1.1758 0.1607 CTC 47838959 CCC 37290873 CAC 42634617 CGC 6737724 C 1.0768 0.8393 0.9596 0.1517 C CTA 36671812 CCA 52352507 CAA 53776608 CGA 6251611 A 0.8254 1.1784 1.2104 0.1407 CTG 57598215 CCG 7815619 CAG 57544367 CGG 7815677 G FIRST NUCLEOTIDE 1.2964 1.1759 1.2952 0.1759 THIRD NUCLEOTIDE ATT 71001746 ACT 45731927 AAT 70880610 AGT 45794017 T 1.5981 1.0293 1.5954 1.0307 ATC 37952376 ACC 33024323 AAC 41380831 AGC 39724813 C 0.8542 0.7433 0.9314 0.8941 A ATA 58649060 ACA 57234565 AAA 109143641 AGA 62837294 A 1.3201 0.7433 2.4566 1.4144 ATG 52222957 ACG 7117535 AAG 56701727 AGG 50430220 G 1.1754 0.1602 1.2763 1.1351 GTT 415577671 GCT 39746348 GAT 37990593 GGT 33071650 T 0.9354 0.8946 0.8551 0.7444 GTC 26866216 GCC 33788267 GAC 26820898 GGC 33774099 C 0.6047 0.7605 0.6037 0.7602 G GTA 32292235 GCA 40907730 GAA 56018645 GGA 43853584 A 0.7268 0.9208 1.2609 0.9871 GTG 42755364 GCG 6744112 GAG 47821818 GGG 37333942 G 0.9623 0.1518 1.0764 0.8203

by Perez, he analyzed the codon populations of 20 various species like Eukaryotes, bacteria and viruses. The results showed that 3 parameters (1, 2, and Phi (φ)) define codon populations within their genomes to a precision of 99% and often 99.999% [29]. For the human and chimpanzee genomes, codon frequencies are 99.99% correlated [29]. This exemplifies the widespread occurrence of Phi (φ) and Fibonacci series not only within the human genome but in other species as well. Conclusion Fractal geometry lays the foundation to understanding the Figure 7: Dragon curve after 7 folds (a), and after 11 folds (b) [27]. complexity of the shapes in nature. In the exploration of the origins the whole human genome. By modeling of the arrangement of the of life through mathematics, the occurrence of the Golden Ratio, nitrogenous bases, Adenine (A), Guanine (G), Cytosine (C), and Fibonacci Series and the underlying Lucas series are observed in Thymine (T), using both the Universal Genetic Code Table Figure 3 several aspects of life on planet earth and within the cosmos. Although and the Dragon-Fractal paradigm, symmetry in the data was revealed. widely identified in non-biological fields such as architecture and Such symmetry showed two attractors towards values of “1” and that art, it has not been well explored in the human biology. Recent of Phi (φ) 1.618 [25]. According to Chaos theory of dynamic systems, work has begun to explore the understanding of such phenomenon attractors can be thought of ‘magnetic’ points to which the initial state documented at several different scales and systems in the human of a complex system evolves towards in its final conditions. Attractors anatomy and physiology ranging from orthopedics, dentistry, the are important in mathematical modeling and systems because they spiral of the human ear, the cardiovascular system and the human serve as the link between Chaos theory and Fractal geometry [28]. genome. The observance of such a seemingly universal concept begs Perez concludes that the ratios between 3-base pair codons sorted by the question of the origin of life; however, more research needs to be A or G in the second base positions and those by T and C in the second performed to explore its physiological role in biology. Understanding base position tend to an attractor value of “1” [25]. The ratios between its functional role may be a keystone to making quantum advances codons C or G and T or A in the second base position tend to a cluster value of “3(φ)/2” [25]. Furthermore, the distance that separates these in several fields such as artificial intelligence, biomedical engineering two attractors of “1” and “3(φ)/2”, is ½(φ). In a 2009 follow-up study designs, and human regeneration, amongst others.

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References 15. Ward DH. Proportional smile design using the recurring esthetic dental (red) proportion. Dent Clin North Am. 2001; 45: 143-154. 1. Goonatilake S. Toward a Global Science. Indiana University Press. 1998; 126. 16. Di Dio C, Macaluso E, Rizzolatti G. The golden beauty: brain response to classical and renaissance sculptures. PLoS One. 2007; 2: e1201. 2. Shah JA. History of Pingala’s Combinatorics. GanitaBharati. 17. Gombrich EH. Tributes. Interpreters of our cultural tradition. Oxford: Phaidon 3. Dunlap RA. The Golden Ratio and Fibonacci Numbers. World Scientific Press. 1984. Publishing, Singapore. Chapter 6.Lucas number and generalized Fibonacci numbers. 1997. 18. Ramachandran VS. A brief tour of human consciousness. New York: Pearson Education. 2004. 4. Littler JW. On the adaptability of man’s hand (with reference to the equiangular curve). Hand. 1973; 5: 187-191. 19. Mitchison GJ. Phyllotaxis and the fibonacci series. Science. 1977; 196: 270- 275. 5. Gupta A, Rash GS, Somia NN, Wachowiak MP, Jones J, Desoky A. The motion path of the digits. J Hand Surg Am. 1998; 23: 1038-1042. 20. Ashrafian H, Athanasiou T. Fibonacci series and coronary anatomy. Heart Lung Circ. 2011; 20: 483-484. 6. Hamilton R, Dunsmuir RA. Radiographic assessment of the relative lengths of the bones of the fingers of the human hand. J Hand Surg Br. 2002; 27: 21. Yetkin G, Sivri N, Yalta K, Yetkin E. Golden Ratio is beating in our heart. Int J 546-548. Cardiol. 2013; 168: 4926-4927.

7. Hutchison AL, Hutchison RL. Fibonacci, littler, and the hand: a brief review. 22. Mandelbot B. The fractal geometry of nature. WH Freeman and Company. Hand (NY). 2010; 5: 364-368. New York, USA. 1982.

8. Rhodes G, Proffitt F, Grady H, Sumich A. Facial symmetry and the perception 23. Perez JC. Chaos, DNA, and Neuro-computers: A golden link: The hidden of beauty. Psychonomic Bulletin &Review. 1998; 5: 659-669. language of genes, global language and order in the human genome. Speculations in Science and Technology. 1991; 14: 336-346. 9. Lombardi RE. The principles of visual perception and their clinical application to denture esthetics. J Prosthet Dent. 1973; 29: 358-382. 24. Yamagishi ME, Shimabukuro AI. Nucleotide frequencies in human genome and fibonacci numbers. Bull Math Biol. 2008; 70: 643-653. 10. Levin E. Dental esthetics and the golden proportion. J Prosthet Dent. 1978; 40: 244-252. 25. Perez JC. Codon populations in single-stranded whole human genome DNA Are fractal and fine-tuned by the Golden Ratio 1.618. Interdiscip Sci. 2010; 11. Ricketts RM. Divine proportion in facial esthetics. Clin Plast Surg. 1982; 9: 2: 228-240. 401-422. 26. Crilly AJ, Earnshaw RA, Jones H. Fractals and Chaos, Springer-Verlag. New 12. Ricketts RM. The biologic significance of the divine proportion and Fibonacci York, USA. 1991. series. Am J Orthod. 1982; 81: 351-370. 27. Chang A, Zhang T. The fractal geometry of the boundary of the dragon 13. Mahshid M, Khoshvaghti A, Varshosaz M, Vallaei N. Evaluation of “golden curves. Journal of recreational mathematics. 2000; 30: 9-22. proportion” in individuals with an esthetic smile. J Esthet Restor Dent. 2004; 16: 185-192. 28. Robert LVT, Advised by Dr. John David, “Attractors: Nonstrange to Chaotic”.

14. Gillen RJ, Schwartz RS, Hilton TJ, Evans DB. An analysis of selected 29. Perez JC. Codex Biogenesis. Resurgence, Li`ege Belgium. 2009. normative tooth proportions. Int J Prosthodont. 1994; 7: 410-417.

Austin J Surg - Volume 2 Issue 5 - 2015 Citation: Persaud-Sharma D and O’Leary JP. Fibonacci Series, Golden Proportions, and the Human Biology. ISSN : 2381-9030 | www.austinpublishinggroup.com Austin J Surg. 2015;2(5): 1066. Persaud-Sharma et al. © All rights are reserved

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